| L(s) = 1 | + 0.539i·2-s − i·3-s + 1.70·4-s + (2.17 − 0.539i)5-s + 0.539·6-s + i·7-s + 2i·8-s − 9-s + (0.290 + 1.17i)10-s + 11-s − 1.70i·12-s + 0.829i·13-s − 0.539·14-s + (−0.539 − 2.17i)15-s + 2.34·16-s + 1.63i·17-s + ⋯ |
| L(s) = 1 | + 0.381i·2-s − 0.577i·3-s + 0.854·4-s + (0.970 − 0.241i)5-s + 0.220·6-s + 0.377i·7-s + 0.707i·8-s − 0.333·9-s + (0.0919 + 0.370i)10-s + 0.301·11-s − 0.493i·12-s + 0.230i·13-s − 0.144·14-s + (−0.139 − 0.560i)15-s + 0.585·16-s + 0.395i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.487660468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.487660468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.17 + 0.539i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.539iT - 2T^{2} \) |
| 13 | \( 1 - 0.829iT - 13T^{2} \) |
| 17 | \( 1 - 1.63iT - 17T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 - 4.70iT - 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 + 1.51iT - 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 5.43iT - 43T^{2} \) |
| 47 | \( 1 + 6.72iT - 47T^{2} \) |
| 53 | \( 1 + 2.26iT - 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 - 0.248T + 71T^{2} \) |
| 73 | \( 1 - 2.92iT - 73T^{2} \) |
| 79 | \( 1 - 0.0422T + 79T^{2} \) |
| 83 | \( 1 - 6.70iT - 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.788467443282788401358565424260, −8.872292340576738514711535415085, −8.149191893039671428393073105426, −7.13026269409953165964557935571, −6.53587033307949755680122161218, −5.74326596862747148669915131216, −5.10329400530480610491740856563, −3.39322449384621660198858184592, −2.24106036446187571263682011711, −1.47085574397744101443805261769,
1.26459095171238992658129628780, 2.54386603915423355011414240412, 3.28556152065115910334424377970, 4.52344768586642535481788896677, 5.56869608695417365132985603458, 6.44829238566624382152632059080, 7.06558776933866896447951568991, 8.191528398986580390264612110347, 9.252330888864473252117907554538, 9.970088461499216378145161158929
